Innumeracy

So I just saw an article about Justin Bieber’s newly released song “U Smile 800% Slower.” This song was purportedly obtained by slowing down Bieber’s original “U Smile” song by 800%. Read that last sentence again and see if you can make sense of it. That’s what I thought. Slowing something down by 800% is gibberish, unless you allow for going in reverse.

To be fair, what they mean is that the new version has been “stretched out” to be 8 times longer than the old version. Equivalently, its new speed is one eighth its old speed. But this is not the same as being slowed down by 800%. In fact, if your new speed is one eighth your old speed, then you’ve slowed down by 87.5%, not 800%, a fact which I’ll prove below.

To understand what’s going on, first consider increasing a quantity by a certain percentage. What does it mean to increase something up by 50%? Well, it’s new value should be 50% more than its old value. For example, if you increase 100 by 50%, what’s the new value? Answer: 150. What did you do? You added 50% of 100 to itself. More generally, if is the quantity we’re increasing by percent, its new value is . Thus, to increase 100 by 75% is to multiply it by , so its new value is 175, right? Right.

Now consider reducing something by a certain percentage. Say you’re driving down the highway at 100 mph. If you slow down by 50%, how fast are you going? Answer: 50 mph. If you slow down instead by 75% how fast are you going? Answer: 25 mph. What did you do? Answer: you subtracted the given percentage of the original quantity from itself. Generally, if is the quantity to be reduced by percent, its new value is . In other words, to find the new (reduced) value, you multiply the original value by . In the 75% case for example, the new value of is . Similarly, if you slow down by 100% how fast are you going? Answer: 0 mph, since . You’ve stopped.

Now, if you slow down by 800% how fast are you going? Answer: You’re going in reverse at 700 mph, since .

In any case, the new song is 1/8th as fast as the original song. If is the speed of the original song, its new speed is . By what percentage was the song slowed down? According to the formula developed above, we need to find a such that . Solving for gives . That is, the song was slowed down by 87.5%, not 800%.